Lecture Notes: Variance Inflating Factor and Multicollinearity

Variance Inflating Factor (VIF)

• Definition: VIF is defined as ( \frac{1}{1-R^2_j} ).
• Implication: If ( R^2_j ) approaches 1, the variance of ( \beta_j ) tends to infinity.
• Consequence: High linear dependence among explanatory variables can lead ( R^2_j ) to be high, implying inflated variance for ( \beta_j ).
• Graphical Representation: Plotting ( R^2 ) vs variance of ( \beta_j ) shows an increasing trend.

Effects of Increased Variance

• Standard Error Increase: As variance of ( \beta_j ) increases, so does the standard error.
• T-statistic Impact: The t-statistic ( \frac{\beta_j}{\text{SE of } \beta_j} ) will decrease, leading to statistically insignificant t-values.
• Multicollinearity Consequence: High ( R^2 ) but insignificant individual t-statistics.

Multicollinearity

• Model Example: ( Y_i = \alpha + \beta_1 X_{1i} + \beta_2 X_{2i} + \mu_i )
• Symptoms: High ( R^2 ) with insignificant t-statistics for explanatory variables.

Detecting Multicollinearity

• Pairwise Correlation:

  • Calculate pairwise correlations between explanatory variables.
  • ( r > 0.8 ) indicates potential multicollinearity.
  • Limitations: Low pairwise correlation does not rule out multicollinearity.

• VIF Method:

  • Calculate VIF for each variable.
  • VIF > 10 suggests multicollinearity.

• Observation:

  • Insignificant individual t-statistics with high ( R^2 ).

Solutions to Multicollinearity

• Perspective: Multicollinearity is a sample problem.
• Modify Sample:
  • Ensure sample includes individuals with varied income and wealth levels.
  • Increase sample size to enhance variability.
• Mathematical Impact: Increasing sample size increases ( \sum (X_j - \bar{X_j})^2 ), reducing variance.

Conclusion

• Further Measures: Will be discussed in the next class, including additional methods to solve multicollinearity.